TY - JOUR

T1 - The Widom-Dyson constant for the gap probability in random matrix theory

AU - Deift, P.

AU - Its, A.

AU - Krasovsky, I.

AU - Zhou, X.

N1 - Funding Information:
Percy Deift was supported in part by NSF Grants # DMS-0296084 and # DMS 0500923. Alexander Its was supported in part by NSF Grants # DMS-0099812 and # DMS-0401009. Xin Zhou was supported in part by NSF Grant # DMS-0071398.

PY - 2007/5/1

Y1 - 2007/5/1

N2 - In the bulk scaling limit for the Gaussian Unitary Ensemble in random matrix theory, the probability that there are no eigenvalues in the interval (0, 2 s) is given by Ps = det (I - Ks), where Ks is the trace-class operator with kernel Ks (x, y) = frac(sin (x - y), π (x - y)) acting on L2 (0, 2 s). In the analysis of the asymptotic behavior of Ps as s → ∞, there is particular interest in the constant term known as the Widom-Dyson constant. We present a new derivation of this constant, which can be adapted to calculate similar critical constants in other problems arising in random matrix theory.

AB - In the bulk scaling limit for the Gaussian Unitary Ensemble in random matrix theory, the probability that there are no eigenvalues in the interval (0, 2 s) is given by Ps = det (I - Ks), where Ks is the trace-class operator with kernel Ks (x, y) = frac(sin (x - y), π (x - y)) acting on L2 (0, 2 s). In the analysis of the asymptotic behavior of Ps as s → ∞, there is particular interest in the constant term known as the Widom-Dyson constant. We present a new derivation of this constant, which can be adapted to calculate similar critical constants in other problems arising in random matrix theory.

KW - Asymptotic expansions

KW - Correlation functions

KW - Random matrices

KW - Riemann-Hilbert problem

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U2 - 10.1016/j.cam.2005.12.040

DO - 10.1016/j.cam.2005.12.040

M3 - Article

AN - SCOPUS:33847058828

VL - 202

SP - 26

EP - 47

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1 SPECIAL ISSUE

ER -